using the dimensionless quantity \(x \equiv \sqrt{2}\sqrt{\frac{\alpha }{c_\mathrm {s,0}}}(r-r_0)\) . The solution is a Parabolic Cylinder Function \(U\)  [1]}
\(p = U\left(-\frac{1}{2}(1+\frac{c_\mathrm {s,0}}{\alpha }k_{r,0}^2),~ x\right).\) 
